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Understanding LTI Systems in Analog and Digital Communications

Explore modulation, sampling, Nyquist theorem, response of LTI systems, random variables, probability functions, and more in analog and digital communications. Learn about convolution, time invariance, eigenfunctions, and characteristics of LTI systems. Develop insights into digital signal processing concepts.

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Understanding LTI Systems in Analog and Digital Communications

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  1. LTI Systems, Probability Analog and Digital Communications Autumn 2005-2006 CS477: Analog and Digital Communications

  2. Sampling • First consider modulation • Product with Cosine in time domain • Convolution with two impulses in frequency domain • Next consider sampling • Product with a train of impulses in time domain • Convolution with a train of impulses in the frequency domain • Nyquist sampling theorem • A bandlimited signal [-B, +B] can be characterized by its samples taken every 1/(2B) seconds. i.e., 2B samples per second • Undersampling leads to aliasing CS477: Analog and Digital Communications

  3. LTI Systems • Linearity If then • Time invariance • Linearity and Time invariance CS477: Analog and Digital Communications

  4. Response of LTI Systems Impulse response: CS477: Analog and Digital Communications

  5. Exponentials and LTI Systems Exponentials are eigenfunctions of LTI systems! LTI Systems can not generate new frequencies! CS477: Analog and Digital Communications

  6. Hilbert Transformer • A filter introducing a constant delay of 90 degrees to the input signal • Hilbert transform does not change the domain; It’s merely a convolution CS477: Analog and Digital Communications

  7. Random Variables • Outcomes and sample space • Random Variables • Mapping outcomes to: • Discrete numbers  Discrete RVs • Real line  Continuous RVs • Cumulative distribution function • One variable • Joint cdf CS477: Analog and Digital Communications

  8. Random Variables • Probability mass function (discrete RV) • Probability density function (cont. RV) • Joint pdf of independent RVs • Mean • Variance • Characteristic function • (IFT of pdf) CS477: Analog and Digital Communications

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